The Stinespring Dilation

We motivated channels as "couple to an environment, evolve unitarily, discard the environment." The Stinespring dilation theorem makes that picture exact and universal: every channel arises this way. Noise is never fundamental — it is unitary evolution on a larger system, seen from the inside.

The theorem

Let $\mathcal{E}:\mathcal{L}(\mathcal{H}_A)\to\mathcal{L}(\mathcal{H}_B)$ be CPTP. Then there is an environment Hilbert space $\mathcal{H}_E$ and a linear isometry $V:\mathcal{H}_A\to\mathcal{H}_B \otimes\mathcal{H}_E$ (so $V^\dagger V = \mathbb{1}_A$ ) such that

\boxed{\;\mathcal{E}(\rho) = \operatorname{tr}_E\!\big(V\rho V^\dagger\big).\;}

One may take $\dim\mathcal{H}_E$ equal to the Kraus rank of $\mathcal{E}$ , and the minimal such dilation is unique up to an isometry on $E$ . Conversely, any map of this form is automatically CPTP — isometric embedding (CP, trace preserving on its image) followed by partial trace (CP, trace preserving) is CPTP. So "CPTP" and "isometry-then-discard" are the same class.

From Kraus operators to the isometry

The dilation is built directly from a Kraus set $\{K_k\}_{k=0}^{r-1}$ . Introduce an environment of dimension $r$ with orthonormal basis $\{|k\rangle_E\}$ and define

V = \sum_{k=0}^{r-1} K_k \otimes |k\rangle_E, \qquad\text{i.e.}\qquad V|\psi\rangle = \sum_k (K_k|\psi\rangle)\otimes|k\rangle_E.

This $V$ is an isometry exactly because of the completeness relation:

V^\dagger V = \sum_{j,k} K_j^\dagger K_k\,\langle j|k\rangle = \sum_k K_k^\dagger K_k = \mathbb{1}.

Tracing out $E$ recovers the channel — the $|k\rangle_E$ basis vectors are orthogonal, so the cross terms vanish:

\operatorname{tr}_E\big(V\rho V^\dagger\big) = \sum_{j,k} K_k\rho K_j^\dagger\,\langle j|k\rangle = \sum_k K_k\rho K_k^\dagger = \mathcal{E}(\rho).

So the abstract environment is concrete: it is a register that records which Kraus branch occurred, and "discarding" it is forgetting that record.

Unitary form

An isometry into $\mathcal{H}_B\otimes\mathcal{H}_E$ can always be extended to a unitary $U$ by adjoining the environment as an input prepared in a fixed state $|0\rangle_E$ , $V|\psi\rangle = U(|\psi\rangle\otimes|0\rangle_E)$ . Then

\mathcal{E}(\rho) = \operatorname{tr}_E\!\big[\,U(\rho\otimes|0\rangle\langle0|_E)U^\dagger\,\big],

recovering the open-systems picture from the module's first lesson, now as a theorem rather than a motivating example.

The complementary channel

The dilation hands you a second channel for free. Tracing out $B$ instead of $E$ defines the complementary channel

\mathcal{E}^c(\rho) = \operatorname{tr}_B\!\big(V\rho V^\dagger\big),

which describes the information that leaks into the environment. The pair $(\mathcal{E}, \mathcal{E}^c)$ are two views of the same isometry, and their interplay is the heart of quantum information theory: a channel transmits well exactly when its complement transmits poorly. This information–disturbance tradeoff underlies the no-cloning theorem, quantum capacities, and the security of quantum key distribution — a channel that lets an eavesdropper (the environment) learn a lot necessarily degrades the legitimate output.

The takeaway

Stinespring's theorem says every CPTP map is an isometry $V$ into a system-plus-environment space followed by discarding the environment, with the environment dimension equal to the Kraus rank and $V=\sum_k K_k\otimes|k\rangle_E$ built straight from the Kraus operators. Equivalently it is a unitary on system-plus-environment with the environment traced out. Tracing out the other factor gives the complementary channel, encoding the information–disturbance tradeoff.